Is it possible to "dissolve" a driver and commit its effects on an animated armature?
Figured it out, with the help of Rich Sedman.

What I'm doing now is establishing a complete animation in Blender, with the use of drivers and empties. Blender drivers are python-dependent, so of course they don't export; but by doing the following I can make the FBX animation-ready.


Bake the movement of the empty. (This option is in the 3D view, under Object->Animation->Bake Action)
Bake the animation of the target mesh in the same way.
Open up the graph view, go to Drivers and then the driver pane on the right, and remove the driver. It's no longer necessary.
Delete the empty that was in use. I suggest this because otherwise, the animation applying to the empty will still export; and FBX doesn't do much for identifying which meshes are associated with which animations. Tools like Unreal and Unity will get confused, and most likely apply it to the end mesh.
Export the animation as an FBX file, and it should be ready to use.